Dynamic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 236

¼ fð U w _ _ u Þ: (9.2)

The variation in fluid content z may then be defined in terms of w :

z ¼r w : (9.3)

is the variation of the fluid volume per
unit volume of the porous material; it is defined as the difference between the strain
of the pore space and the strain of the fluid volume in the pore space and is
dimensionless (see ( 8.16 )).
In addition to the slight changes in kinematic notation noted above, Biot ( 1962a , b )
employed slightly different notation for the poroelastic constitutive relations. As an
introduction to these changes consider the inverse of ( 8.17 ),

Recall that the variation in fluid content z

ðz A EÞ; p ¼ M ð p ¼ M ðz A ij E ij ÞÞ;

where M is the inverse of defined by ( 8.18 ),

L

M ¼ 1 = L (9.4)

and the representation of stress T as a function of the strain E and the variation in
fluid content z , rather than as a function of the strain and the pore pressure p ,is

¼fC d

T

ðA A Þg E MA

C ijkm þ

þ M z; ð T ij ¼ð MA km A ij Þ E km MA ij zÞ:

(9.5)

Biot ( 1962a , equations ( 9.11 )) employed a slightly different notation for the two
previous expressions, namely

¼ Z d

T

E z M

Z ijkm E km z

; ð T ij ¼ M ij Þ (9.6)

and

¼M E þ z p M ; ð p ¼ M km E km þ z M Þ; (9.7)

M and Z d are related to C eff , A and C d above by

where Biot's parameters M ,

¼ C d

Z d

M ¼

MA; ðA AÞ; ð M ij ¼ MA ij Þ; þ M

(9.8)

Z ijkm ¼ C ijkm þ ð MA ij A km Þ: